Corollary

Let A be a commutative Banach algebra. The space $\mathsf{X}'(A)$ is compact. The space $\mathsf{X}(A)$ is locally compact, and is compact if A admits a unit element.

Let $A'$ be the dual of the Banach space A. Its unit ball $A'_1$ is weakly compact (EVT, III, p. 17, cor. 2). By Theorem 1, one has $\mathsf{X}'(A)\subset A'_1$. Moreover, $\mathsf{X}'(A)$ is closed in $A'$ for the weak topology, because it is the intersection of the weakly closed sets

$X_{x,y}=\{f\in A'|f(xy)-f(x)f(y) = 0\}$ (for $x,y\in A$).

It follows that $\mathsf{X}'(A)$ is compact and that $\mathsf{X}(A) =\mathsf{X}'(A)-\{0\}$ is locally compact.

If A admits a unit element 1, then $\mathsf{X}(A)$ is the set of $\chi \in$ $\mathsf{X}'(A)$ such that $\chi (1) = 1$, and is therefore a closed subset, and hence compact, of $\mathsf{X}'(A)$.

Every compact space is homeomorphic to $\mathsf{X}(A)$ for a suitable commutative unital Banach algebra A (cf. I, p. 32, cor. 2).